Let
\begin{equation*}%\label{Fall-Factorial-Dfn-Eq}
\langle\alpha\rangle_n=
\prod_{k=0}^{n-1}(\alpha-k)=
\begin{cases}
\alpha(\alpha-1)\dotsm(\alpha-n+1), & n\ge1\\
1,& n=0
\end{cases}
\end{equation*}
and
\begin{equation*}
(\alpha)_n=\prod_{\ell=0}^{n-1}(\alpha+\ell)
=
\begin{cases}
\alpha(\alpha+1)\dotsm(\alpha+n-1), & n\ge1\\
1, & n=0
\end{cases}
\end{equation*}
denote the falling and rising factorials of $\alpha\in\mathbb{C}$ respectively.

In Remark 3.1 of the paper [1] below, the formula
\begin{equation}\label{Bell-1-lambda}\tag{1}
B_{n,k}\Biggl(1, 1-\alpha, (1-\alpha)(1-2\alpha),\dotsc, \prod_{\ell=0}^{n-k}(1-\ell\alpha)\Biggr)
=\frac{(-1)^k}{k!} \sum_{\ell=0}^k (-1)^{\ell} \binom{k}{\ell} \prod_{q=0}^{n-1}(\ell-q\alpha)
\end{equation}
for $\alpha\in\mathbb{C}$ and $n\ge k\ge0$ was concluded.

In Theorem 2.1 of the paper [2] below, the formulas
\begin{equation}\label{Bell-fall-Eq}\tag{2}
B_{n,k}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-k+1})
=\frac{(-1)^k}{k!}\sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}\langle\alpha\ell\rangle_n
\end{equation}
and
\begin{equation}\label{Bell-fall-Eq-inv}\tag{3}
\sum_{\ell=0}^{k}\frac{B_{n,\ell}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-\ell+1})}{(k-\ell)!}
=\frac{\langle\alpha k\rangle_n}{k!}
\end{equation}
for $n\ge k\ge0$ and $\alpha\in\mathbb{R}$ were established.

As consequnces of the formulas \eqref{Bell-fall-Eq} and \eqref{Bell-fall-Eq-inv}, the formulas
\begin{equation}\label{Bell-rise-Eq}\tag{4}
B_{n,k}((\alpha)_1, (\alpha)_2, \dotsc, (\alpha)_{n-k+1})
=\frac{(-1)^k}{k!}\sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}(\alpha\ell)_n
\end{equation}
and
\begin{equation}\label{Bell-rise-Eq-inv}\tag{5}
\sum_{\ell=0}^{k}\frac{B_{n,\ell}((\alpha)_1, (\alpha)_2, \dotsc, (\alpha)_{n-\ell+1})}{(k-\ell)!}
=\frac{(\alpha k)_n}{k!}
\end{equation}
for $\alpha\in\mathbb{C}$ and $n\ge k\ge0$ were derived in Corollary 2.1 of the paper [2].

The formulas \eqref{Bell-1-lambda} and \eqref{Bell-fall-Eq} were reviewed in Section 1.3 of the article [3] below.

The formulas \eqref{Bell-1-lambda}, \eqref{Bell-fall-Eq}, and \eqref{Bell-rise-Eq} are equivalent to
\begin{equation}\tag{6}
B_{n,k}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-k+1})
=\sum _{j=k}^n s(n,j)\alpha^jS(j,k)
\end{equation}
for $n\ge k\ge0$ and $\alpha\in\mathbb{R}$ at https://mathoverflow.net/a/88071/147732, where $s(n,j)$ and $S(j,k)$ stand for the Stirling numbers of the first and second kinds respectively.

References
 1. B.-N. Guo and F. Qi, *Viewing some ordinary differential equations from the angle of derivative polynomials*, Iran. J. Math. Sci. Inform. **16** (2021), no. 1, 77--95; available online at https://doi.org/10.29252/ijmsi.16.1.77.
 2. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, *Closed formulas and identities for the Bell polynomials and falling factorials*, Contributions to Discrete Mathematics **15** (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111.
 3. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, *Special values of the Bell polynomials of the second kind for some sequences and functions*, Journal of Mathematical Analysis and Applications **491** (2020), no. 2, Paper No.124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.